munoz (gm7794) – HW12 – radin – (54915)
1
This
printout
should
have
19
questions.
Multiplechoice questions may continue on
the next column or page – find all choices
before answering.
001
10.0 points
Rewrite the series
4
parenleftbigg
1
4
parenrightbigg
2
sin 3
5

4
parenleftbigg
1
4
parenrightbigg
3
sin 4
6
+ 4
parenleftbigg
1
4
parenrightbigg
4
sin 5
7
+
. . .
using summation notation.
1.
sum =
∞
summationdisplay
k
=3
parenleftbigg

1
4
parenrightbigg
k
−
1
4 sin
k
k
+ 2
correct
2.
sum =
25
summationdisplay
k
=3
parenleftbigg

1
4
parenrightbigg
k
−
1
4 sin
k
k
+ 2
3.
sum =
∞
summationdisplay
k
=1
parenleftbigg

1
4
parenrightbigg
k
4 sin(
k
+ 2)
k
+ 4
4.
sum =
80
summationdisplay
k
=2
parenleftbigg
1
4
parenrightbigg
k
4 sin(
k
+ 1)
k
+ 3
5.
sum =
∞
summationdisplay
k
=3
parenleftbigg

1
4
parenrightbigg
k
−
1
4 sin
k
k
+ 1
Explanation:
The given series is an infinite series, so
two of the answers must be incorrect because
they are finite series written in summation
notation.
Starting summation at
k
= 3 we see that
the general term of the infinite series is
a
k
= 4
parenleftbigg

1
4
parenrightbigg
k
−
1
sin
k
k
+ 2
.
Consequently,
sum =
∞
summationdisplay
k
=3
parenleftbigg

1
4
parenrightbigg
k
−
1
4 sin
k
k
+ 2
.
002
10.0 points
Determine whether the series
∞
summationdisplay
n
=0
2 (cos
nπ
)
parenleftbigg
2
3
parenrightbigg
n
is convergent or divergent, and if convergent,
find its sum.
1.
convergent with sum

5
6
2.
convergent with sum

6
5
3.
convergent with sum
6
5
correct
4.
convergent with sum 6
5.
convergent with sum

6
6.
divergent
Explanation:
Since
cos
nπ
= (

1)
n
,
the given series can be rewritten as an infinite
geometric series
∞
summationdisplay
n
=0
2
parenleftbigg

2
3
parenrightbigg
n
=
∞
summationdisplay
n
=0
a r
n
in which
a
= 2
,
r
=

2
3
.
But the series
∑
∞
n
=0
ar
n
is
(i) convergent with sum
a
1

r
when

r

<
1,
and
(ii) divergent when

r
 ≥
1.
Consequently, the given series is
convergent with sum
6
5
.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
munoz (gm7794) – HW12 – radin – (54915)
2
003
10.0 points
Determine if the series
∞
summationdisplay
k
=1
5 + 2
k
3
k
converges or diverges, and if it converges, find
its sum.
1.
converges with sum =
9
2
correct
2.
converges with sum = 3
3.
converges with sum = 5
4.
converges with sum =
7
2
5.
series diverges
6.
converges with sum = 4
Explanation:
An infinite geometric series
∑
∞
n
=1
a r
n
−
1
(i) converges when

r

<
1 and has
sum =
a
1

r
,
while it
(ii) diverges when

r
 ≥
1
.
Now
∞
summationdisplay
k
=1
5
3
k
=
∞
summationdisplay
k
=1
5
3
parenleftBig
1
3
parenrightBig
k
−
1
is a geometric series with
a
=
r
=
1
3
<
1.
Thus it converges with
sum =
5
2
,
while
∞
summationdisplay
k
=1
2
k
3
k
=
∞
summationdisplay
k
=1
2
3
parenleftBig
2
3
parenrightBig
k
−
1
is a geometric series with
a
=
r
=
2
3
<
1.
Thus it too converges, and it has
sum = 2
.
Consequently, being the sum of two conver
gent series, the given series
converges with sum =
5
2
+ 2 =
9
2
.
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '08
 Cepparo
 Calculus, Mathematical Series, Munoz

Click to edit the document details